Math Doubts

What is a Surd?

Definition

A quantity that expresses an unresolved root of a number is called a surd.

Introduction

The irrational numbers are one type of real numbers, whereas there are some special forms of irrational numbers and they express the unresolved roots of positive integers. So, they are called the surds.

what is a surd?

Let’s start the concept of a surd in mathematics to know what a surd really is.

Explanation

The square root of two is an irrational number and its arithmetic value is given below.

$\sqrt{2}$ $\,=\,$ $1.4142135623\cdots$

The number of decimal places is infinite because it is an unresolved root of a number. So, its value cannot be evaluated exactly in mathematics and it is called a surd.

Now, let’s see some more examples to clearly know what a surd is in mathematics.

Examples

$\sqrt{3}$, $\sqrt{4}$ and $\sqrt{5}$

The values of square roots of $3$, $4$ and $5$ are given here.

$(1).\,\,$ $\sqrt{3}$ $\,=\,$ $1.7320508075\cdots$

$(2).\,\,$ $\sqrt{4}$ $\,=\,$ $2$

$(3).\,\,$ $\sqrt{5}$ $\,=\,$ $2.2360679774\cdots$

Now, let’s discuss about above irrational form real numbers.

  1. The number of decimal places is infinite, in the case of both the square root of $3$ and the square root of $5$, because the square roots of $3$ and $5$ are unsolvable. So, $\sqrt{3}$ is a surd and $\sqrt{5}$ is also a surd.
  2. The square root of $4$ is $2$ because it is solvable. So, $\sqrt{4}$ is not a surd.

Let’s see some more examples of a surd.

$(1).\,\,$ $\sqrt[\Large 3]{2}, \, \sqrt[\Large 3]{7}, \, \sqrt[\Large 3]{10}$ and so on.

$(2).\,\,$ $\sqrt[\Large 4]{3}, \, \sqrt[\Large 4]{5}, \, \sqrt[\Large 4]{6}$ and so on.

$(3).\,\,$ $\sqrt[\Large 5]{2}, \, \sqrt[\Large 5]{4}, \, \sqrt[\Large 5]{9}$ and so on.

The degree of a root can be any positive integer and every unresolved root of a positive real number is a surd. A surd is also a radical but remember, every radical is not a surd.

Notation

When $a$ denotes a positive rational number and $n$ denotes a positive integer, the $n$th root of a expresses an unresolved root of a number and it is called a surd.

In general, a surd is written as $\sqrt[\Large n]{a}$ algebraically in mathematics.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved